Injections de Sobolev probabilistes et applications (original) (raw)

In this article, we give probabilistic versions of Sobolev embeddings on any Riemannian manifold (M,g)(M,g)(M,g). More precisely, we prove that for natural probability measures on L2(M)L^2(M)L2(M), almost every function belong to all spaces Lp(M)L^p(M)Lp(M), p<+inftyp<+\inftyp<+infty. We then give applications to the study of the growth of the LpL^pLp norms of spherical harmonics on spheres mathbbSd\mathbb{S}^dmathbbSd: we prove (again for natural probability measures) that almost every Hilbert base of L2(mathbbSd)L^2(\mathbb{S}^d)L2(mathbbSd) made of spherical harmonics has all its elements uniformly bounded in all Lp(mathbbSd),p<+inftyL^p(\mathbb{S}^d), p<+\inftyLp(mathbbSd),p<+infty spaces. We also prove similar results on tori mathbbTd\mathbb{T}^dmathbbTd. We give then an application to the study of the decay rate of damped wave equations in a frame-work where the geometric control property on Bardos-Lebeau-Rauch is not satisfied. Assuming that it is violated for a measure 0 set of trajectories, we prove that there exists almost surely a rate. Finally, we conclude with an application to the study of the H1H^1H1-supercritical wave equation, for which we prove that for almost all initial data, the weak solutions are strong and unique, locally in time.

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